= Solution
Use the <time-harmonic wave> convention $\psi e^{-i\omega t}$ and write $\mathbf z=(y,z)$, $E(0,\mathbf z)=E_0(\mathbf z)$, with a deterministic incident <wave envelope>. The printed speed ratio is inconsistent with $k=k_0n$: the <refractive index> used below is $n=c_0/c$. Substituting $\psi=e^{ik_0x}E$ in the <Helmholtz equation> gives
$$
2ik_0\partial_xE+\partial_x^2E+\Delta_\perp E+k_0^2(n^2-1)E=0.
$$
The <paraxial approximation> discards $\partial_x^2E$. The usual weak-fluctuation model also linearizes $n^2-1=2\mu W+O(\mu^2W^2)$, giving
$$
\partial_xE=(D+S_x)E,\qquad D=\frac{i}{2k_0}\Delta_\perp,\qquad S_x=ik_0\mu W(x,\mathbf z).
$$
Dropping the quadratic contrast is an additional weak-fluctuation assumption, not a consequence of small propagation angles. Although the linearized random potential generates attenuation of order $\mu^2$, it does not retain every effect of that order in the literal finite-correlation index: the discarded quadratic contrast can also produce a mean <wave phase> shift. This linearization must precede a <Gaussian white noise> limit: the square of ideal <white noise> has no ordinary pointwise meaning.
The <split-step Fourier method> alternates <free-space diffraction>, $U_h=e^{hD}$, with a <random phase screen>,
$$
E(x+h,\mathbf z)\simeq U_h\left[e^{i\phi_h(\mathbf z)}E(x,\mathbf z)\right],\qquad
\phi_h(\mathbf z)=k_0\mu\int_x^{x+h}W(s,\mathbf z)\,ds.
$$
For jointly <Gaussian random fields>, <Gaussian phase averaging> gives $\mathbb E e^{i\phi_h}=e^{-\operatorname{Var}(\phi_h)/2}$. More generally, a product of $p$ fields and $q$ conjugate fields picks up $\exp(i\sum_a\epsilon_a\phi_h(\mathbf z_a))$, with signs $\epsilon_a=+1$ or $-1$. Its screen average is determined entirely by the <wave phase> <covariance matrix>. The deterministic <diffraction> step acts on each coordinate, with opposite signs on factors formed by <complex conjugation>. This is the basis of the field-moment equations.
There is an important closure qualification. A <stationary Gaussian random field> need not have independent longitudinal increments. For finite-correlation fluctuations, the unlinearized <parabolic wave equation> instead gives
$$
\partial_xm=Dm+ik_0\mu\,\mathbb E[WE]+\frac{ik_0\mu^2}{2}\mathbb E[W^2E].
$$
In the weak-fluctuation model, the exact first-moment equation is
$$
\partial_xm=Dm+ik_0\mu\,\mathbb E[WE],\qquad m=\mathbb E E.
$$
The last term cannot in general be replaced by a constant times $m$. An exact generic solution of the linearized model is
$$
m(x)=\mathbb E\left[\mathcal T\exp\left(\int_0^x(D+ik_0\mu W(s,\cdot))\,ds\right)\right]E_0,
$$
where $\mathcal T$ denotes <time ordering> of the propagation operators. For the unlinearized model, add $ik_0\mu^2W^2/2$ inside the propagation generator. The <covariance function> of the medium is needed to evaluate this expression; One-point <Gaussian distributions> alone would not even determine the joint <wave phase> statistics.
For example, omit <diffraction> and take a longitudinal <autocovariance function> $R(s)=\mathbb E[W(s,\mathbf z)W(0,\mathbf z)]$. Direct <Gaussian phase averaging> gives
$$
m(x,\mathbf z)=E_0(\mathbf z)\exp\left[-k_0^2\mu^2\int_0^x(x-s)R(s)\,ds\right],\qquad
\partial_xm=-k_0^2\mu^2\left(\int_0^xR(s)\,ds\right)m.
$$
Even here, unit-<variance> <stationary Gaussian random fields> with different $R$ give different answers. In the general problem the <diffraction> and multiplication operators do not commute, so this scalar attenuation cannot simply be multiplied by $U_x$ without an additional approximation.
The standard closed answer uses the <Markov approximation for a random medium>, made explicit in part ii. Let the longitudinally integrated <covariance kernel> be
$$
B(\mathbf r)=\int_{\mathbb R}\mathbb E[W(s,\mathbf z+\mathbf r)W(0,\mathbf z)]\,ds.
$$
Replace the medium by longitudinal <Gaussian white noise> with this strength. A screen of thickness $h$ then has $\mathbb E[\phi_h(\mathbf z_a)\phi_h(\mathbf z_b)]=k_0^2\mu^2hB(\mathbf z_a-\mathbf z_b)$ and is independent of the incoming field. For the moment $M=\mathbb E\prod_a E_a^{(\epsilon_a)}$, expanding both steps to order $h$ gives
$$
\partial_xM=\frac{i}{2k_0}\sum_a\epsilon_a\Delta_{\mathbf z_a}M
-\frac{k_0^2\mu^2}{2}\sum_{a,b}\epsilon_a\epsilon_bB(\mathbf z_a-\mathbf z_b)M.
$$
In particular, the <coherent attenuation in a white-noise random medium> and its <Fresnel propagator> solution are
$$
\partial_xm=\frac{i}{2k_0}\Delta_\perp m-\gamma m,\qquad\gamma=\frac{k_0^2\mu^2B(0)}2.
$$
$$
\boxed{m(x)=e^{-\gamma x}U_xE_0.}
$$
Equivalently, write $W\,dx=d\beta_x$, where $\beta_x(\mathbf z)$ is <Brownian motion> in $x$ with transverse <covariance kernel> $\mathbb E[d\beta_x(\mathbf z)d\beta_x(\mathbf z')]=B(\mathbf z-\mathbf z')dx$. The <Stratonovich integral> formulation is $dE=DE\,dx+ik_0\mu E\circ d\beta_x$. Its <Itô integral> form is
$$
dE=(DE-\gamma E)\,dx+ik_0\mu E\,d\beta_x.
$$
The mean of the <Itô integral> vanishes, independently confirming the attenuation drift. For $x>0$ in two transverse dimensions,
$$
(U_xE_0)(\mathbf z)=\frac{k_0}{2\pi i x}\int_{\mathbb R^2}\exp\left(\frac{ik_0|\mathbf z-\mathbf z'|^2}{2x}\right)E_0(\mathbf z')\,d\mathbf z'.
$$
Thus a unit <plane wave> has $m=e^{-\gamma x}$. For a general incident <wave envelope>, the <Fresnel propagator> supplies its spreading. The attenuation is redistribution between the <coherent and diffuse wave fields>, rather than <wave absorption>: for a field and its conjugate at the same point, the screen contribution in the second-moment equation cancels.
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